1. solve the equation x = (2x+ 3)1/2
2. consider f1(x) = ln(x + 1) + ln (x-1) and f2(x) = ln(x^2-1).
a) state domains and ranges of f1 and f2.
b) sketch the curves y = f1(x) and y = f2(x).

1.  ; 2. Domain:    Range: all real numbers

Step-by-step explanation:

Let's find the solutions.

1. Solve the equation so:

Although we have two answers, remember that from the original equation the result of is never negative. So -1 do not solve the equation.

In conlcusion, the equation is solved by x=3.

2A. Domains and ranges of f1(x) and f2(x)

Using logarithmic property we have:

because:

is defined by   then:

so the domain of f1(x) is

Now for the range:

notice that is always positive, so the range of f1(x) is all real numbers.

Be aware that although point number two of the problem mentioned two equations, f1(x)=f2(x) by logarithmic properties, so their domains and ranges are the same.

2B. Graph of f1(x) is attached. Because f1(x)=f2(x) both functions plot equal.

reduce the friction between its moving parts

step-by-step explanation:

in fact, by reducing the friction between the moving parts of the machine, it is possible to reduce the energy wasted due to this friction; therefore, more input energy is converted into useful work, and this will improve the efficiency of the machine.

attachment?

step-by-step explanation: